Langevin dynamcs MCMC

August 17, 2020 — September 5, 2024

Bayes
estimator distribution
functional analysis
Markov processes
Monte Carlo
neural nets
optimization
probabilistic algorithms
probability
SDEs
stochastic processes
Figure 1: Randomly exploring the posterior space.

Sampling using the particular SDE that is the Langevin equation. It can be rather close to optimising, via SDE representations of SGD.

1 Langevin dynamics

2 Metropolis-adjusted Langevin algorithm (MALA)

3 Continuous time

See log concave distributions for a family of distributions where this works especially well because implcit (more nearly continuous-time exact) solutions are available Hodgkinson, Salomone, and Roosta (2019).

Left-field, Max Raginsky, Sampling Using Diffusion Processes, from Langevin to Schrödinger:

the Langevin process gives only approximate samples from \(\mu\). I would like to discuss an alternative approach that uses diffusion processes to obtain exact samples in finite time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the Schrödinger bridge problem.

4 Annealed

TBC Jolicoeur-Martineau et al. (2022);Song and Ermon (2020a);Song and Ermon (2020b).

Yang Song, Generative Modeling by Estimating Gradients of the Data Distribution

5 Incoming

Holden Lee, Andrej Risteski introduce the connection between log-concavity and convex optimisation.

\[ x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I). \]

6 References

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